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We study the geometry of projective manifolds whose tangent bundles are nefon sufficiently general curves (i.e. the tangent bundle is generically nef) andshow that manifolds whose anticanonical bundles are semi-ample have thisproperty. Furthermore we introduce a notion of sufficient nefness andinvestigate the relation with manifolds whose anticanonical bundles are nef.

Varieties with generically nef tangent bundles